SOME PROPERTIES OF COMBINATION OF SOLUTIONS TO SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH ANALYTIC COEFFICIENTS OF [P; Q]-ORDER IN THE UNIT DISC

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SOME PROPERTIES OF COMBINATION OF SOLUTIONS TO SECOND-ORDER

LINEAR DIFFERENTIAL EQUATIONS WITH ANALYTIC COEFFICIENTS OF

[P;Q]-ORDER IN THE UNIT DISC

Article  in  International Journal of Applied Mathematics and Computer Science · January 2014 CITATION

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SOME PROPERTIES OF COMBINATION OF SOLUTIONS TO SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH ANALYTIC COEFFICIENTS OF

[P; Q]-ORDER IN THE UNIT DISC

BENHARRAT BELAÏDI1 _{AND ZINELÂABIDINE LATREUCH}

Abstract. In this paper, we consider some properties on the growth and oscillation of combination of solutions of the linear dierential equation

f00_{+ A (z) f}0_{+ B (z) f = 0;}

with analytic coecients A (z) and B (z) with [p; q] order in the unit disc
= fz 2 C : jzj < 1g.

1. Introduction and preliminaries In the year 2000, Heittokangas rstly investigated the growth and oscillation theory of complex dierential equation

(1.1) f(k)_{+ A}

k 1(z) f(k 1)+


+ A0(z) f = 0;

where A0(z) ;


; Ak 1(z) are analytic functions in

the unit disc (see, [15]). It is well-known that all solu-tions of (1.1) are analytic funcsolu-tions (see, [15]). After him many authors (see, [4], [5], [8], [9], [10], [11], [12], [13], [16], [22]) have investigated the complex dier-ential equation (1.1) and the second-order dierdier-ential equations

(1.2) f00_{+ A (z) f}0_{+ B(z)f = 0;}

(1.3) f00_{+ A (z) f = 0}

with analytic and meromorphic coecients in the unit disc
: In ([17], [18]), Juneja and his co-authors investigated some properties of entire functions of [p; q] order, and obtained some results concerning their growth. Later, Liu, Tu and Shi; Xu, Tu and Xuan; Li and Cao; Belaïdi; Latreuch and Belaïdi applied the concepts of entire (meromorphic) func-tions in the complex plane and analytic funcfunc-tions in the unit disc
= fz 2 C : jzj < 1g of [p; q] order to investigate the complex dierential equation (1.1) (see [6], [7], [22], [23], [24], [26]). In this paper, we will

use this concept to study the growth and the oscilla-tion of the combinaoscilla-tion of two linearly independent

solutions f1and f2 of equation (1.2) in the unit disc.

In this paper, we assume that the reader is famil-iar with the fundamental results and the standard notations of the Nevanlinna's theory on the complex plane and in the unit disc
= fz 2 C : jzj < 1g ; see ([14], [15], [19], [20], [25]).

In the following, we will give similar denitions as in ([17], [18]) for analytic and meromorphic func-tions of [p; q]-order, [p; q]-type and [p; q]-exponent of convergence of the zero-sequence in the unit disc. Denition 1.1. ([6],[22] ) Let p  q  1 be inte-gers, and let f be a meromorphic function in
; the [p; q]-order of f (z) is dened by

[p;q](f) = lim sup r!1 log+_p T (r; f) log_q 1 1 r ;

where T (r; f) is the Nevanlinna characteristic function of f: For an analytic function f in
, we also dene M;[p;q](f) = lim sup r!1 log+p+1M (r; f) log_q 1 1 r ; where M (r; f) = max jzj=rjf (z)j :

Remark 1.1. It is easy to see that 0  [p;q](f) 

+1 (0  M;[p;q](f)  +1 ); for any p 

q  1: By Denition 1.1, we have that [1;1] =

1_{corresponding author}

2010 Mathematics Subject Classication. 34M10, 30D35.

Key words and phrases. Analytic functions, linear dierential equations, [p; q] order, [p; q] exponent of convergence of the sequence of distinct zeros, unit disc.

12 B. BELAÏDI AND Z. LATREUCH

 (f) (M;[1;1] = M(f)) and [2;1] = 2(f)

M;[2;1]= M;2(f)
.

For the relationship between [p;q](f) and

M;[p;q](f) we have the following double inequality.

Proposition 1.1. ([6]) Let p  q  1 be integers, and let f be an analytic function in
of [p; q]-order.

(i) If p = q  1; then

[p;q](f)  M;[p;q](f)  [p;q](f) + 1:

(ii) If p > q  1, then

[p;q](f) = M;[p;q](f) :

Denition 1.2. ([22] ) Let p  q  1 be integers. The [p; q]-type of a meromorphic function f (z) in
of [p; q]-order  (0 <  < +1 ) is dened by [p;q](f) = lim sup r!1 log+ p 1T (r; f)  log_{q 1} 1 1 r :

Denition 1.3. ([22]) Let p  q  1 be inte-gers. The [p; q]-exponent of convergence of the zero-sequence of f (z) in
is dened by [p;q](f) = lim sup r!1 log+_p Nr;1 f  log_q 1 1 r ; where Nr;1 f 

is the integrated counting function of zeros of f (z) in fz : jzj  rg. Similarly, the [p; q]-exponent of convergence of the sequence of distinct zeros of f (z) in
is dened by

[p;q](f) = lim sup r!1 log+_p Nr;1 f  log_q 1 1 r ; where Nr;1 f 

is the integrated counting function of distinct zeros of f (z) in fz : jzj  rg.

The study of the properties of linearly indepen-dent solutions of complex dierential equations is an old problem. In ([2], [3]), Bank and Laine obtained

some results about the product E = f1f2 of two

lin-early independent solutions f1 and f2 of (1.3) in the

complex plane. In [21], the authors have investigated the relations between the polynomial of solutions of (1.2) and small functions in the complex plane. They

showed that w = d1f1+ d2f2keeps the same

proper-ties of growth and oscillation of fj (j = 1; 2) ; where

f1 and f2 are two linearly independent solutions of

(1.2) and obtained the following results.

Theorem 1.1. ([21]) Let A (z) and B (z) be entire functions of nite order such that  (A) <  (B) and  (A) <  (B) < +1 if  (B) =  (A) >

0. Let dj(z) (j = 1; 2) be entire functions

that are not all vanishing identically such that

maxf (d1) ;  (d2)g <  (B). If f1 and f2 are two

nontrivial linearly independent solutions of (1.2),

then the polynomial of solutions w = d1f1+ d2f2

satises

 (w) =  (f1) =  (f2) = +1

and

2(w) =  (B) :

In the same paper, the authors studied also the zeros of the dierence between the polynomial of

so-lutions w = d1f1+ d2f2and entire functions of nite

order.

The remainder of the paper is organized as fol-lows. In Section 2, we shall show our main results which improve and extend many results in the above-mentioned papers. Section 3 is for some lemmas and basic theorems. The other sections are for the proofs of our main results.

2. Main results

A natural question arises: What can be said about similar situations in the unit disc
for equation (1.2) in the terms of [p; q] order? Before we state our re-sults, we dene h and (z) by

h =     H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 H16    ; where H1= d1; H2= 0; H3= d2; H4= 0; H5= d01; H6= d1; H7= d02; H8= d2; H9= d001 d1B; H10= 2d01 d1A; H11= d002 d2B; H12= 2d02 d2A; H13= d1(3) 3d01B +d1AB d1B0; H14= 3d001 2d01A d1B + d1A2 d1A0; H15= d(3)2 3d02B + d2AB d2B0; H16= 3d002 2d02A d2B + d2A2 d2A0 and (z) = 2 d1d2d02 d22d01
h '(3)+ 2'00+ 1'0+ 0';

where ' 6 0, dj(j = 1; 2) are analytic functions of

nite [p; q]-order in
and

(2.1) 2= 2 d1d2d 0 2 d22d01
A 3d1d2d002+ 3d22d001 h ; 1= 6d2(d 0 1d002 d02d001) + 2d2(d1d02 d2d01) B h (2.2) +2d2(d1d02 d2d01) A0+ 3d2(d2d001 d1d002) A h ;

0 = _h1[(d1d02d002 3d2d02d001+ 2d2d01d002) A + 4d1(d02)2+ 3d22d001 3d1d2d002 4d2d01d02
B +2 d2d01d02 d1(d02)2
A0 +2 d1d2d02 d22d01
B0_{+ 6(d}0 2)2d001 2d1d02d0002 + 2d2d01d0002 (2.3) 3d2d001d002 6d01d02d002+ 3d1(d002)2]:

Theorem 2.1. Let p  q  1 be integers, and let A (z) and B (z) be analytic functions in
of

nite [p; q]-order such that [p;q](A) < [p;q](B)

and 0 < [p;q](A) < [p;q](B) < +1 if [p;q](B) =

[p;q](A) > 0. Let dj(z) (j = 1; 2) be analytic

func-tions that are not all vanishing identically such

that maxf[p;q](d1) ; [p;q](d2)g < [p;q](B). If f1

and f2 are two nontrivial linearly independent

so-lutions of (1.2), then the polynomial of soso-lutions

(2.4) w = d1f1+ d2f2

satises

[p;q](w) = [p;q](f1) = [p;q](f2) = +1

and

[p;q](B)  [p+1;q](w)  M;

where and in the following M =

maxM;[p;q](A) ; M;[p;q](B) : Furthermore, if

p > q  1; then

[p+1;q](w) = [p;q](B) :

Example 2.1. ([16]) For  > 0; the

func-tions f1(z) = exp(exp (1 z) 
) and f2(z) =

exp (1 z) 
_{exp(exp (1 z) 
) are linearly}

independent solutions of (1.2) satisfying

[1;1](f1) = [1;1](f2) = +1 and [2;1](f1) = [2;1](f2) = ; where A (z) = 2 exp (1 z) 
(1 z)+1  (1 z)+1 1 +  1 z and B (z) = 2exp 2(1 z) 
(1 z)2+2 :

It is clear that [1;1](A) = [1;1](B) and [1;1](A) <

[1;1](B) : Then, by Theorem 2.1 for any two

analytic functions di(z) (i = 1; 2) of nite order

[1;1](di) < +1 (i = 1; 2) that are not all vanishing

identically such that max[1;1](d1) ; [1;1](d2) <

[1;1](B), the combination w = d1f1+ d2f2 is of

innite order [1;1](w) = +1 and [2;1](w) = .

From Theorem 2.1, we can obtain the following result.

Corollary 2.1. Let p  q  1 be integers, and let

fi(z) (i = 1; 2) be two nontrivial linearly

indepen-dent solutions of (1.2), where A (z) and B (z) 6 0 are analytic functions of nite [p; q] order in

such that [p;q](A) < [p;q](B) or [p;q](A) =

[p;q](B) > 0 and 0 < [p;q](A) < [p;q](B) < +1;

and let dj(z) (j = 1; 2; 3) be analytic functions in

satisfying

max[p;q](dj) : j = 1; 2; 3 < [p;q](B)

and

d2(z) f2+ d1(z) f1= d3(z) :

Then dj(z)  0 (j = 1; 2; 3) :

Theorem 2.2. Under the hypotheses of Theorem 2.1, let ' (z) 6 0 be an analytic function in

with nite [p; q] order such that (z) 6 0: If f1

and f2 are two nontrivial linearly independent

so-lutions of (1.2), then the polynomial of soso-lutions

w = d1f1+ d2f2 satises (2.5) [p;q](w ') = [p;q](w ') = [p;q](w) = +1 and [p;q](B)  [p+1;q](w ') = (2.6) [p+1;q](w ') = [p+1;q](w)  M: Furthermore, if p > q  1; then [p+1;q](w ') = [p+1;q](w ') (2.7) = [p+1;q](w) = [p;q](B) :

Theorem 2.3. Let p  q  1 be integers, and let A (z) and B (z) be analytic functions in
of

nite [p; q] order such that [p;q](A) < [p;q](B).

Let dj(z) ; bj(z) (j = 1; 2) be nite [p; q] order

analytic functions in
such that d1(z) b2(z)

d2(z) b1(z) 6 0. If f1 and f2 are two nontrivial

linearly independent solutions of (1.2), then

[p;q]  d1f1+ d2f2 b1f1+ b2f2  = +1 and [p;q](B)  [p+1;q]  d1f1+ d2f2 b1f1+ b2f2   M: Furthermore, if p > q  1; then [p+1;q]  d1f1+ d2f2 b1f1+ b2f2  = [p;q](B) :

14 B. BELAÏDI AND Z. LATREUCH

3. Auxiliary lemmas

Lemma 3.1. ([14], [15], [25]) Let f be a meromor-phic function in the unit disc and let k 2 N: Then

m 

r;f_f(k) 

= S (r; f) ;

where S (r; f) = Olog+_{T (r; f) + log} 1

1 r

 ;

pos-sibly outside a set E1 [0; 1) withR_{E1 1 r}dr < +1 :

Lemma 3.2. ([1], [15]) Let g : (0; 1) ! R and h : (0; 1) ! R be monotone increasing functions such that g (r)  h (r) holds outside of an

excep-tional set E2  [0; 1) for which R_{E2 1 r}dr < +1 .

Then there exists a constant d 2 (0; 1) such that if s (r) = 1 d (1 r) ; then g (r)  h (s (r)) for all r 2 [0; 1):

Lemma 3.3. ([6]) Let p  q  1 be integers.

If A0(z) ;


; Ak 1(z) are analytic functions of

[p; q] order in the unit disc
; then every solu-tion f 6 0 of (1.1) satises

[p+1;q](f) = M;[p+1;q](f)

 maxM;[p;q](Aj) : j = 0; 1;


; k 1 :

Lemma 3.4. ([22]) Let p  q  1 be integers. Let

Aj (j = 0;


; k 1) ; F 6 0 be analytic functions

in
; and let f (z) be a solution of the dierential equation f(k)_{+ A} k 1(z) f(k 1)+


+ A1(z) f0+ A0(z) f = F satisfying max[p;q](Aj) (j = 0;


; k 1) ; [p;q](F ) < [p;q](f) =   +1: Then we have [p;q](f) = [p;q](f) = [p;q](f) and [p+1;q](f) = [p+1;q](f) = [p+1;q](f) :

Lemma 3.5. ([22]) Let p  q  1 be integers, and let f and g be non-constant meromorphic func-tions of [p; q]-order in
: Then we have

[p;q](f + g)  max[p;q](f) ; [p;q](g)

and

[p;q](fg)  max[p;q](f) ; [p;q](g) :

Furthermore, if [p;q](f) > [p;q](g) ; then we

ob-tain

[p;q](f + g) = [p;q](fg) = [p;q](f) :

Lemma 3.6. ([22]) Let p  q  1 be integers, and let f and g be meromorphic functions of [p;

q]-order in
such that 0 < [p;q](f) ; [p;q](g) <

+1 and 0 < [p;q](f) ; [p;q](g) < +1: Then, we have (i) If [p;q](f) > [p;q](g) ; then [p;q](f + g) = [p;q](fg) = [p;q](f) : (ii) If [p;q](f) = [p;q](g) and [p;q](f) 6= [p;q](g) ; then [p;q](f + g) = [p;q](fg) = [p;q](f) = [p;q](g) :

Lemma 3.7. ([22]) Let p  q  1 be integers, and

let Aj(z) (j = 0;


; k 1) be analytic functions

in
satisfying max[p;q](Aj) : j = 1;


; k 1 < [p;q](A0) : If f 6 0 is a solution of (1.1), then [p;q](f) = +1 and [p;q](A0)  [p+1;q](f)  maxM;[p;q](Aj) : j = 0;


; k 1 : Furthermore, if p > q  1; then [p+1;q](f) = [p;q](A0) :

Lemma 3.8. Let p  q  1 be integers, and let A (z) and B (z) be analytic functions in
of

-nite [p; q] order such [p;q](A) < [p;q](B). If f1

and f2 are two nontrivial linearly independent

so-lutions of (1.2), then f1 f2 is of innite [p; q]-order and [p;q](B)  [p+1;q]  f1 f2   M: Furthermore, if p > q  1; then [p+1;q]  f1 f2  = [p;q](B) :

Proof. Suppose that f1 and f2 are two

nontriv-ial linearly independent solutions of (1.2). Since

[p;q](B) > [p;q](A), then by Lemma 3.7

[p;q](f1) = [p;q](f2) = +1; [p;q](B) 

(3.1) [p+1;q](f1) = [p+1;q](f2)  M:

Furthermore, if p > q  1; then

[p+1;q](f1) = [p+1;q](f2) = [p;q](B) :

On the other hand (3.2)  f1 f2 0 = W (f_f1₂; f2) 2 ;

where W (f1; f2) = f1f20 f2f10 is the Wronskian of

f1and f2: By using (1.2) we obtain that

W0_(f

1; f2) = A (z) W (f1; f2);

which implies that

(3.3) W (f1; f2) = K exp(

Z

where RA(z)dz is the primitive of A (z) and K 2 Cn f0g : By (3.2) and (3.3) we have (3.4)  f1 f2 0 = Kexp( R A(z)dz) f2 2 : Since [p;q](f2) = +1; [p+1;q](f2)  [p;q](B) > [p;q](A) if p  q  1 and [p+1;q](f2) = [p;q](B) >

[p;q](A) if p > q  1; then by using (3.1) and Lemma

3.5 we obtain from (3.4) [p;q]  f1 f2  = [p;q](f2) = +1; [p;q](B)  [p+1;q]  f1 f2  = [p+1;q](f2)  M if p  q  1 and [p+1;q]  f1 f2  = [p+1;q](f2) = [p;q](B) if p > q  1: 

Lemma 3.9. ([6]) Let p  q  1 be integers. Let f be a meromorphic function in the unit disc

such that [p;q](f) =  < +1, and let k  1 be an

integer. Then for any " > 0; m  r;f_f(k)  = O  exp_{p 1}  ( + ") log_q  1 1 r 

holds for all r outside a set E3  [0; 1) with

R

E3

dr

1 r < +1 :

Lemma 3.10. Let p  q  1 be integers, and let f be a meromorphic function in
with [p; q]

-order 0 < [p;q](f) =  < +1 and [p; q] - type

0 < [p;q](f) =  < +1: Then for any given  < ;

there exists a subset E4 of [0; 1) that has an

in-nite logarithmic measure R_E4 dr

1 r = +1 such

that log_{p 1}T (r; f) > hlog_{q 1} 1

1 r

i_

holds for

all r 2 E4:

Proof. By the denitions of [p; q] order and [p; q]

-type, there exists an increasing sequence frmg+1m=1

[0; 1) (rm! 1 ) satisfying _m1 + 1 _m1
rm< rm+1 and lim m!+1 log_{p 1}T (rm; f)  log_{q 1} 1 1 rm _ = :

Then there exists a positive integer m0such that for

all m  m0and for any given 0 < " <  ; we have

(3.5) logp 1T (rm; f) > ( ")  logq 1  1 1 rm _ : For any given  <  "; there exists a positive integer

m1such that for all m  m1we have

(3.6) 2 4logq 1 1 m1
 ₁ 1 r  log_{q 1} 1 1 r  3 5  > _":

Take m  m2 = max fm0; m1g : By (3.5) and (3.6),

for any r 2 [rm;_m1 + 1 _m1
rm]; we have

log_{p 1}T (r; f)  log_{p 1}T (rm; f) > ( ")  logq 1  ₁ 1 rm   ( ")  logq 1  1 _m1   1 1 r _ >   log_{q 1}  1 1 r _ : Set E4 = _m=m+1[ 2  rm;_m1 + 1 _m1
rm; then there holds mlE4= +1_X m=m2 1 m+(1_Zm1)rm rm dt 1 t = X+1 m=m2 log_{m 1}m = +1:  Lemma 3.11. Let p  q  1 be integers, and let A (z) and B (z) be analytic functions in
of

nite [p; q] order such [p;q](A) < [p;q](B) and

0 < [p;q](A) < [p;q](B) < +1 if [p;q](B) =

[p;q](A) > 0. If f 6 0 is a solution of (1.2), then

[p;q](f) = +1 ; [p;q](B)  [p+1;q](f)  M

and

[p+1;q](f) = [p;q](B)

if p > q  1.

Proof. If [p;q](B) > [p;q](A), then the result can be

deduced by Lemma 3.7. We prove only the case when

[p;q](B) = [p;q](A) =  and [p;q](B) > [p;q](A) >

0: Since f 6 0; then by (1.2) B =  f00 f + A f0 f  :

Suppose that f is of nite [p; q] order [p;q](f) =

 < +1: Then by Lemma 3.9 T (r; B)  T (r; A)+O  exp_{p 1}  ( + ") log_q  1 1 r 

holds for all r outside a set E3 [0; 1) withR_{E3 1 r}dr <

+1; which implies by using Lemma 3.2 the contra-diction

[p;q](B)  [p;q](A) :

Hence [p;q](f) = +1: By Lemma 3.3, we have

[p+1;q](f) = M;[p+1;q](f)

 maxM;[p;q](A) ; M;[p;q](B) :

On the other hand, since [p;q](f) = +1; then by

Lemma 3.1 (3.7) T (r; B)  T (r; A)+O  log+_{T (r; f) + log} 1 1 r 

16 B. BELAÏDI AND Z. LATREUCH

holds for all r outside a set E1  [0; 1) with

R

E1

dr

1 r < +1. By [p;q](B) > [p;q](A) > 0, we

choose 0; 1 satisfying [p;q](B) > 0 > 1 >

[p;q](A) such that for r ! 1 ; we have

(3.8) T (r; A)  exp_{p 1}  1  log_{q 1}  1 1 r  :

By Lemma 3.10, there exists a subset E4 [0; 1) of

innite logarithmic measure such that

(3.9) T (r; B) > exp_{p 1}  0  log_{q 1}  1 1 r  :

By (3.7)-(3.9) we obtain for all r 2 E4nE1

exp_{p 1}  0  log_{q 1}  1 1 r   exp_{p 1}  1  log_{q 1}  1 1 r  (3.10) +O  log+_{T (r; f) + log} 1 1 r  : By using (3.10) and Lemma 3.2, we obtain

[p;q](B)  [p+1;q](f) : Hence [p;q](B)  [p+1;q](f)  maxM;[p;q](A) ; M;[p;q](B) if p  q  1 and [p+1;q](f) = [p;q](B) if p > q  1:  4. Proof of Theorem 2.1

Proof. In the case when d1(z)  0 or d2(z)  0; then

the conclusions of Theorem 2.1 are trivial. Suppose

that f1and f2are two nontrivial linearly independent

solutions of (1.2) and dj(z) 6 0 (j = 1; 2) : Then by

Lemma 3.11, we have

[p;q](f) = +1; [p;q](B)  [p+1;q](f)  M

if p  q  1 and

[p+1;q](f) = [p;q](B)

if p > q  1. Suppose that d1 = cd2; where c is a

complex number: Then, we obtain

w = d1f1+ d2f2= cd2f1+ d2f2= (cf1+ f2) d2:

Since f = cf1 + f2 is a solution of (1.2) and

[p;q](d2) < [p;q](B) ; then we have

[p;q](w) = [p;q](cf1+ f2) = +1;

[p;q](B)  [p+1;q](w) = [p+1;q](cf1+ f2)  M

if p  q  1 and

[p+1;q](w) = [p+1;q](cf1+ f2) = [p;q](B)

if p > q  1. Suppose now that d16 cd2 where c is a

complex number. Dierentiating both sides of (2.4), we obtain

(4.1) w0_{= d}0

1f1+ d1f10 + d02f2+ d2f20:

Dierentiating both sides of (4.1), we obtain

(4.2) w00_{= d}00 1f1+2d01f10+d1f100+d002f2+2d02f20+d2f200: Substituting f00 j = A (z) fj0 B (z) fj(j = 1; 2) into equation (4.2), we have w00_{= (d}00 1 d1B) f1+ (2d01 d1A) f10 (4.3) + (d00 2 d2B) f2+ (2d02 d2A) f20:

Dierentiating both sides of (4.3) and by substituting f00 j = A (z) fj0 B (z) fj (j = 1; 2) ; we obtain w000₌_d(3) 1 3d01B + d1(AB B0)  f1 + 3d00 1 2d01A + d1 A2 A0 B

f10 +d(3)₂ 3d0 2B + d2(AB B0)  f2 (4.4) + 3d00 2 2d02A + d2 A2 A0 B

f20: By (2.4) and (4.1)-(4.4) we have 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : w = d1f1+ d2f2; w0_{= d}0 1f1+ d1f10+ d02f2+ d2f20; w00_{= (d}00 1 d1B) f1+ (2d01 d1A) f10 + (d00 2 d2B) f2+ (2d02 d2A) f20; w000₌_d(3) 1 3d01B + d1(AB B0)  f1 + 3d00 1 2d01A + d1 A2 A0 B

f10 +d(3)₂ 3d0 2B + d2(AB B0)  f2 + 3d00 2 2d02A + d2 A2 A0 B

f20:

To solve this system of equations, we need rst to prove that h 6 0: By simple calculations we obtain

h =     H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 H16     = 2 (d1d02 d2d01)2B + d2 2d01d001+ d12d02d002 d1d2d01d002 d1d2d02d001
A 2 (d1d02 d2d01)2A0+ 2d1d2d01d0002 + 2d1d2d02d0001 6d1d2d001d002 6d1d01d02d002 6d2d01d02d001 +6d1(d02)2d001+ 6d2(d10)2d002 2d22d01d0001 2d2 1d02d0002 + 3d21(d200)2+ 3d22(d001)2:

It is clear that (d1d02 d2d01)26 0 because d16= cd2.

Since

and (d1d02 d2d01)2 6 0; then by using Lemma 3.6

we can deduce that [p;q](h) = [p;q](B) > 0: Hence

h 6 0: By Cramer's method we have

f1=     w H2 H3 H4 w0 _{H6 H7 H8} w00 _H 10 H10 H12 w(3) _{H14 H15 H16}     h (4.5) = 2 d1d2d02 d22d01
h w(3)+2w00+1w0+0w;

where j (j = 0; 1; 2) are meromorphic functions

in
of nite [p; q]-order which are dened in

(2.1)-(2.3). By (4.5) and Lemma 3.5, we have

[p;q](f1)  [p;q](w) ([p+1;q](f1)  [p+1;q](w)) and

by (2.4) we have [p;q](w)  [p;q](f1) ([p+1;q](w) 

[p+1;q](f1)): Thus [p;q](w) = [p;q](f1) and

[p+1;q](w) = [p+1;q](f1) : 

5. Proof of Corollary 2.1

Proof. We suppose there exists j = 1; 2; 3 such

that dj(z) 6 0 and we obtain a contradiction. If

d1(z) 6 0 or d2(z) 6 0; then by Theorem 2.1 we have

[p;q](d1f1+ d2f2) = +1 = [p;q](d3) < [p;q](B)

which is a contradiction. Now if d1(z)  0,

d2(z)  0 and d3(z) 6 0 we obtain also a

contra-diction. Hence dj(z)  0 (j = 1; 2; 3). 

6. Proof of Theorem 2.2 Proof. By Theorem 2.1, we have

[p;q](w) = +1; [p;q](B)  [p+1;q](w)  M

if p  q  1 and

[p+1;q](w) = [p;q](B)

if p > q  1. Set g (z) = d1f1 + d2f2 ':

Since [p;q](') < +1; then by Lemma 3.5 we

have [p;q](g) = [p;q](w) = +1; [p+1;q](g) = [p+1;q](w) : In order to prove [p;q](w ') = [p;q](w ') = +1 and [p+1;q](w ') = [p+1;q](w ') = [p+1;q](w) ; we need to prove only [p;q](g) = [p;q](g) = +1 and [p+1;q](g) = [p+1;q](g) = [p+1;q](w) : By w = g + ' we get from (4.5) (6.1) f1= 2 d1d2d 0 2 d22d01
h g(3)+ 2g00+ 1g0+ 0g + ; where = 2 d1d2d02 d22d01
h '(3)+ 2'00+ 1'0+ 0':

Substituting (6.1) into equation (1.2), we obtain 2 d1d2d02 d22d01
h g(5)+ 4 X j=0 jg(j) = ( 00_{+ A (z)} 0_{+ B (z) ) = F (z) ;}

where j(j = 0;


; 4) are meromorphic functions of

nite [p; q]-order in
. Since 6 0 and [p;q]( ) <

+1; it follows that is not a solution of (1.2), which implies that F (z) 6 0: Then, by applying Lemma 3.4

we obtain (2.5), (2.6) and (2.7). 

7. Proof of Theorem 2.3

Proof. Suppose that f1and f2are two nontrivial

lin-early independent solutions of (1.2). Then by Lemma 3.8, we have (7.1) [p;q]  f1 f2  = +1; [p;q](B)  [p+1;q]  f1 f2   M if p  q  1 and (7.2) [p+1;q] _f 1 f2  = [p;q](B) if p > q  1. Set g = f1 f2: Then w (z) = d_b1(z) f1(z) + d2(z) f2(z) 1(z) f1(z) + b1(z) f2(z) =d_b1(z) g (z) + d2(z) 1(z) g (z) + b2(z):

By Lemma 3.5, it follows that

[p+1;q](w)

 maxf[p+1;q](dj) ; [p+1;q](bj) ( j = 1; 2) ; [p+1;q](g)g

(7.3) = [p+1;q](g) :

On the other hand

g (z) = b_b2(z) w (z) d2(z)

1(z) w (z) d1(z);

which implies by Lemma 3.5 that [p;q](w) 

[p;q](g) = +1 and [p+1;q](g)  maxf[p+1;q](dj) ; [p+1;q](bj) ( j = 1; 2) ; [p+1;q](w)g (7.4) = [p+1;q](w) : By using (7.1)-(7.4), we obtain [p;q](w) = [p;q](g) = +1; [p;q](B)  [p+1;q](w) = [p+1;q](g)  M if p  q  1 and [p+1;q](w) = [p+1;q](g) = [p;q](B) if p > q  1. 

18 B. BELAÏDI AND Z. LATREUCH

Acknowledgment

The authors are grateful to the referee for his/her valuable comments which lead to the improvement of this paper.

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Laboratory of Pure and Applied Mathematics University of Mostaganem (UMAB)

B. P. 227 Mostaganem, Algeria E-mail address: belaidi@univ-mosta.dz Department of Mathematics

Laboratory of Pure and Applied Mathematics University of Mostaganem (UMAB)

B. P. 227 Mostaganem, Algeria E-mail address: z.latreuch@gmail.com

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